Postulates of Special Relativity

Special relativity replaces Galilean (Newtonian) spacetime once one demands that the laws of electromagnetism — in particular the speed of light — be the same in every inertial frame. This page sets up the historical and operational motivation, states the postulates precisely, and derives the single most important consequence: the relativity of simultaneity. The transformations themselves are derived in Lorentz transformations; the geometric arena is built in Spacetime and the interval.

We work in inertial frames throughout and adopt natural units $c = 1$ where convenient, restoring factors of $c$ when physically illuminating. The metric convention is mostly-minus, $η_{μν} = diag (+ 1, - 1, - 1, - 1)$ , matching the QFT section.

The crisis in classical physics

By 1900 two pillars of physics were mutually inconsistent:

Newtonian mechanics is invariant under the Galilean group: velocities add linearly, $u^{'} = u - v$ , and there is a universal absolute time $t^{'} = t$ .
Maxwell's electromagnetism predicts electromagnetic waves traveling at a fixed speed $c = 1/ μ_{0} ε_{0}$ . This speed is built into the field equations with no reference to any source or observer.

If velocities add Galileanly, $c$ could hold in only one preferred frame — the rest frame of the hypothetical luminiferous ether. Maxwell's equations would then be frame-dependent, contradicting the apparent universality of electromagnetism.

Michelson–Morley

The 1887 Michelson–Morley interferometer searched for the Earth's motion through the ether by comparing light travel times along perpendicular arms. The expected fringe shift $\sim (v / c)^{2}$ was absent: no ether wind was detectable. Lorentz and FitzGerald proposed an ad hoc longitudinal contraction to save the ether; Einstein's 1905 move was instead to drop the ether and elevate the constancy of $c$ to a postulate.

The two postulates

Einstein's 1905 framework rests on two statements:

Principle of relativity. The laws of physics take the same form in all inertial frames. No experiment singles out a preferred inertial frame or detects absolute uniform motion.
Invariance of the speed of light. Light propagates in vacuum with the same speed $c$ in every inertial frame, independent of the motion of the source or observer.

The first postulate generalizes Galileo's principle to all physics (including electromagnetism), not just mechanics. The second is the radical one: combined with the first, it forces simultaneity, time intervals, and lengths to become frame-dependent.

Tacit assumptions: homogeneity and isotropy

Two background assumptions are taken for granted whenever one derives the Lorentz transformations: spacetime is homogeneous (no event is privileged — the laws are the same everywhere and everywhen) and space is isotropic (no direction is privileged). These are not separate postulates so much as part of what "inertial frame" means; homogeneity is what forces the frame transformations to be linear (free particles map to free particles), and isotropy is what makes the boost depend only on the speed $∣ v ∣$ , not the direction. They are usually folded into the principle of relativity rather than listed alone, but every derivation step labeled "homogeneity" or "isotropy" appeals to them.

Operational definitions: clocks and rods

Relativity is built on operational definitions of measurement:

An inertial frame is a global rigid lattice of synchronized clocks at rest, on which a free particle moves at constant velocity.
Einstein synchronization. Two clocks at $A$ and $B$ are synchronized by sending a light pulse from $A$ at time $t_{A}$ , reflecting at $B$ , and returning at $t_{A}^{'}$ ; clock $B$ reads $t_{B} = \frac{1}{2} (t_{A} + t_{A}^{'})$ at reflection. This defines simultaneity within a frame and relies on the isotropy of $c$ .
A measurement is always tied to local coincidences of events at lattice points. Frame-dependence of intervals is then a statement about how two such lattices compare.

Relativity of simultaneity

Why simultaneity needs defining

It is tempting to think "two events happen at the same time" is self-evident. But what does it mean for an event here and an event far away to be simultaneous? You cannot stand at both places at once, and any signal you use to compare them — sound, light, a runner — takes time to travel. Einstein's insight (1905) was that simultaneity at a distance is not given by nature; it must be defined by a procedure.

His own examples:

The clock and the train (1905 paper). "If I say the train arrives at 7 o'clock, I mean the pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events." Time at one place is just the reading of a local clock at the coincidence of two events. Comparing a distant clock requires synchronizing it — and the only frame-independent signal available is light, hence Einstein synchronization above.
The lightning and the train embankment (1916 popular book). Two bolts strike the rails at $A$ and $B$ , far apart. An observer midway on the embankment sees the two flashes arrive together and calls them simultaneous. A passenger at the midpoint of a train speeding from $A$ to $B$ moves toward $B$ 's flash and away from $A$ 's, so receives $B$ first — and rightly calls $B$ earlier. Neither is wrong; "simultaneous" carries no meaning until a frame and a synchronization rule are fixed.

So simultaneity is a convention, fixed per frame by Einstein synchronization, not an absolute. The consequence is immediate:

The decisive break from Newtonian intuition: two events simultaneous in one inertial frame are generally not simultaneous in another.

Consider a train moving at speed $v$ ; a flash is emitted at the center. In the train frame, light reaches both ends at once (equal distances, speed $c$ ). In the platform frame, the rear end moves toward the flash and the front away from it, so light reaches the rear first.

Both observers are correct: simultaneity is frame-relative. Quantitatively, two events separated by $Δ x$ and simultaneous ( $Δ t = 0$ ) in one frame are separated in time by

$Δ t^{'} = - γ \frac{v Δ x}{c ^{2}}$

in a frame boosted by $v$ , where $γ = (1 - v^{2} / c^{2})^{- 1/2}$ . This single fact dissolves the twin and ladder paradoxes (see Paradoxes) and underlies time dilation and length contraction.

Simultaneity as the bridge

Although relativity of simultaneity is a consequence of the two postulates, it is also what lets them be stated and coexist:

It makes postulate 2 meaningful. "Light has speed $c$ in every frame" is empty until distant clocks are synchronized — and synchronization is a choice of simultaneity. So a simultaneity convention is logically prior to assigning the speed of light any value; Einstein opens the 1905 paper with this analysis for that reason.
It removes the apparent contradiction. "How can every observer measure the same $c$ ?" seems impossible only while simultaneity is absolute. Once "same time at two places" is frame-dependent, the two postulates stop conflicting — simultaneity is the hinge that makes them jointly consistent.

So simultaneity neither generates nor replaces the postulates, but it is their operational precondition and the conceptual key that lets the constancy of $c$ and the relativity principle hold together.

What survives and what changes

Quantity	Galilean	Special-relativistic
Time	absolute, $t^{'} = t$	frame-dependent
Simultaneity	absolute	relative
Velocity addition	$u^{'} = u - v$	$u^{'} = (u - v) / (1 - uv / c^{2})$
Invariant	time interval	spacetime interval $Δ s^{2}$
Symmetry group	Galilean	Poincaré

The postulates fix the geometry: see Spacetime and the invariant interval for the Minkowski structure, then Lorentz transformations for the explicit frame change. The symmetry group they define is studied in The Lorentz and Poincaré groups, with abstract background in group theory.

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